On the reconstruction via Urysohn-Chlodovsky operators

Harun Karsli

doi:10.2478/gm-2020-0012

.blurhash-client-img { display: none !important; }

On the reconstruction via Urysohn-Chlodovsky operators

General Mathematics

Volume 28 (2020): Issue 2 (December 2020)

By: Harun Karsli

Open Access

|Dec 2020

Abstract

The main first goal of this work is to introduce an Urysohn type Chlodovsky operators defined on positive real axis by using the Urysohn type interpolation of the given function f and bounded on every finite subinterval. The basis used in this construction are the Fréchet and Prenter Density Theorems together with Urysohn type operator values instead of the rational sampling values of the function. Afterwards, we will state some convergence results, which are generalization and extension of the theory of classical interpolation of functions to operators.

References

[1] U. Abel, H. Karsli, Asymptotic expansions for Bernstein-Durrmeyer-Chlodovsky polynomials, Results Math, 73:104, 2018, 12 pages, https://doi.org/10.1007/s00025-018-0863-0.10.1007/s00025-018-0863-0
Search in Google Scholar Back to article
[2] U. Abel, H. Karsli, A complete asymptotic expansion for Bernstein-Chlodovsky polynomials for functions on ℝ,, Mediterranean Journal of Mathematics, 2020 accepted.10.1007/s00009-020-01632-1
Search in Google Scholar Back to article
[3] F. Altomare, M. Campiti, Korovkin-Type Approximation Theory and its Applications, De Gruyter Studies in Mathematics, Walter de Gruyter and Co., Berlin, 17, 1994.10.1515/9783110884586
Search in Google Scholar Back to article
[4] C. Bardaro, J. Musielak, G. Vinti, Nonlinear Integral Operators and Applications, De Gruyter Series in Nonlinear Analysis and Applications, vol. 9, 2003.10.1515/9783110199277
Search in Google Scholar Back to article
[5] P. L. Butzer, On Bernstein Polynomials, Ph.D. Thesis, University of Toronto, 1951.
Search in Google Scholar Back to article
[6] P. L. Butzer, On two dimensional Bernstein polynomials, Canad. J. Math., vol. 5, 1953, 107-113.10.4153/CJM-1953-014-2
Search in Google Scholar Back to article
[7] P. L. Butzer, H. Karsli, Voronovskaya-type theorems for derivatives of the Bernstein-Chlodovsky polynomials and the Szasz-Mirakyan operator, Comment. Math., vol. 49, 2009, 33-58.
Search in Google Scholar Back to article
[8] P. L. Butzer, R. J. Nessel, Fourier Analysis and Approximation, Academic Press, New York, London, 1971.10.1007/978-3-0348-7448-9
Search in Google Scholar Back to article
[9] I. Chlodowsky, Sur le développement des fonctions définies dans un intervalle infini en séries de polynomes de M. S. Bernstein, Compositio Math., vol. 4, 1937, 380-393.
Search in Google Scholar Back to article
[10] I. I. Demkiv, On Approximation of the Urysohn operator by Bernstein type operator polynomials, Visn. L’viv. Univ., Ser. Prykl. Mat. Inform., no. 2, 2000, 26 - 30.
Search in Google Scholar Back to article
[11] H. Karsli, Recent results on Chlodovsky operators, Stud. Univ. Babeş-Bolyai Math., vol. 56, no. 2, 2011, 423-436.
Search in Google Scholar Back to article
[12] H. Karsli, Approximation by Urysohn type Meyer-König and Zeller operators to Urysohn integral operators, Results Math., vol. 72, no. 3, 2017, 1571-1583.10.1007/s00025-017-0729-x
Search in Google Scholar Back to article
[13] H. Karsli, Approximation results for Urysohn type nonlinear Bernstein operators, Advances in Summability and Approximation Theory, Springer, Singapore, 2018, 223-241.10.1007/978-981-13-3077-3_14
Search in Google Scholar Back to article
[14] H. Karsli, Voronovskaya-type theorems for Urysohn type nonlinear Bernstein operators, Math. Methods Appl. Sci., vol. 42, no. 16, 2019, 5190-5198.10.1002/mma.5261
Search in Google Scholar Back to article
[15] H. Karsli, Approximation Results for Urysohn Type Two Dimensional Nonlinear Bernstein Operators, Const. Math. Anal., vol. 1, no. 1, 2018, 45-57.10.33205/cma.453027
Search in Google Scholar Back to article
[16] H. Karsli, Some Approximation Properties of Urysohn type nonlinear operators, Stud. Univ. Babes-Bolyai Math., vol. 64, no. 2, 2019, 183-196.10.24193/subbmath.2019.2.05
Search in Google Scholar Back to article
[17] H. Karsli, On Urysohn type Generalized Sampling Operators, Dolomites Research Notes on Approximation, 2020, accepted.10.3934/mfc.2021015
Search in Google Scholar Back to article
[18] G. G. Lorentz, Bernstein Polynomials, University of Toronto Press, Toronto, 1953.
Search in Google Scholar Back to article
[19] V. L. Makarov, I. I. Demkiv, Approximation of the Urysohn operator by operator polynomials of Stancu type, Ukrainian Math Journal, vol. 64, no. 3, 2012, 356-386.10.1007/s11253-012-0652-y
Search in Google Scholar Back to article
[20] V. L. Makarov, V. V. Khlobystov, Interpolation method for the solution of identification problems for a functional system described by the Urysohn operator, Dokl. Akad. Nauk SSSR, vol. 300, no. 6, 1988, 1332-1336.
Search in Google Scholar Back to article
[21] A. D. Polyanin, A. V. Manzhirov, Handbook of Integral Equations, 2nd ed., Chapman and Hall/CRC Press, 2008.10.1201/9781420010558
Search in Google Scholar Back to article
[22] P. Urysohn, Sur une classe d’equations integrales non lineaires, Mat. Sb., vol. 31, 1923, 236-255.
Search in Google Scholar Back to article
[23] P. Urysohn, On a type of nonlinear integral equation, Mat. Sb., vol. 31, 1924, 236-355.
Search in Google Scholar Back to article
[24] A. M. Wazwaz, Linear and Nonlinear Integral Equations: Methods and Applications, Beijing and Springer-Verlag Berlin, Higher Education press, 2011.
Search in Google Scholar Back to article
[25] P. P. Zabreiko, A. I. Koshelev, M. A. Krasnosel’skii, S. G. Mikhlin, L. S. Rakovscik, V. Ja. Stetsenko, Integral Equations: A Reference Text, Noordhoff Int. Publ., Leyden, 1975.
Search in Google Scholar Back to article